Secure Distributed Matrix Multiplication Under Arbitrary Collusion Pattern

Abstract

We study the secure distributed matrix multiplication (SDMM) problem under arbitrary collusion pattern. In the one-sided SDMM problem, where only one matrix of the matrix multiplication needs to be kept secure, we propose an achievable scheme that attains the optimal normalized download cost. The optimal scheme distributes a different number of encoded copies to each server, and the servers that collude more with others are given fewer encoded copies. The converse result is proved using Shearer’s lemma. In the two-sided SDMM problem under arbitrary collusion pattern, where the user would want to keep both matrices of the matrix multiplication secure, we provide an achievable scheme whose key parameters, including the method with which the random matrices are appended, the number of random matrices appended, the number of encoded copies generated, the number of encoded copies distributed to each server, are given by the proposed algorithm. We also demonstrate, via numerical results, the performance of the proposed scheme in terms of normalized upload-download cost trade-off, and show that it is much better than the current known scheme devised for the homogeneous collusion pattern.